Reducing the degree of Apollonius Diagram Predicates

David Millman

Advisors: Sylvain Pion and Christophe Delage

July 6th 2006

Background

o Voronoi Diagram

o Apollonius Diagram

o Power Diagram

o Incremental Algorithm

o Invert and project

Voronoi Diagram

Given Points pi and m,

Euclidian distance:d p

i,m( ) = pi -m

Given P = p1…pn{ }The Voronoi cell of pi:

V pi( ) = m∈° d ∀j,d pi,m( ) ≤d pj,m( ){ }

Apollonius Diagram

Weighed point or Site si is defined by pi, the point and wi the weight

Given Site si and Point m,

weighted distance:d+ si,m( ) = pi -m -wi

Given S = s1…sn{ }Additively weighted Voronoi cell of si:V si( ) = m∈° d ∀j,d+ si,m( ) ≤d+ s j,m( ){ }

Apollonius Graph

• Dual of Apollonius Diagram

• Vertex is a Site

• Edge two sites whose AW-Voronoi Cell share a face

Power Diagram

Given stie σ i and x, Power of x:

dP σ i,x( ) = x−ci( )2−ri

2

Given Σ = σ1…σn{ }The Power cell σ i:

Ρ σ i( ) = x∈° d ∀j,dP ci,x( ) ≤dP c j,x( ){ }

Weighed point or Site σ i is defined by ci, the center and ri the radius

Incremental Algorithm

The Three basic steps

1. Locate nearest neighbor

2. Check if new site is trivial

3. Update • Vertex Conflict• Edge Conflict

Vertex Conflict

Edge Conflict

Invert and Project

Invert and Project

Invert and Project Convex Hull

Predicates

• PredicatesVertex Conflict, Edge Conflict

• Sub PredicatesOrientation, Radical Side, Radical Intersection,

Power Test, Order On a Line

Orientation

Radical Side

Radical Intersection

Vertex Conflict (revisited)

• There are 6 cases, and we have vertex conflict when:

O-, RI-, RS{+|-}

O+, RI+, RS+O-, RI+, RS+

Vertex Conflict (degeneracies)

• When Orientation is 0

• When Radical Intersection is 0

Pertubations

Consistent ordering

1. Max Radius

2. Lexigraphically

Grow the largest site by +ε

Edge Conflict (revisited)

• We are looking for the case of q breaking 2 into multiple sections, when inverted and projection onto the unit sphere.

Numerical Results

• 66% speedup for vertex conflict 4 finite sites

• 39% speedup for vertex conflict 3 finite and 1 infinite site

• 10-20% lessfilter failures in nearly degenerate cases

Further Work

• Optimize Edge Conflict

• Optimizations of the Incremental Algorithm

• Degeneracies in 3D

• Reduction of Exact Computations

Reducing the degree of Apollonius Diagram Predicates

Merci!!!